Integrated circuits (ICs) are often designed to meet aggressive device density and circuit performance specifications. This trend has resulted in a reduction of the interconnect metal pitch and has increased the number of metallization levels. Aggressive interconnect scaling has increased current densities and associated thermal effects.
Thermal effects are an inherent aspect of electrical power distribution and signal transmission through the interconnects due to self-heating (or Joule heating) caused by resistance to the flow of electrical current. For a given process, circuit designers for IC designs using the process are provided maximum allowable values for three different interconnect current densities. These interconnect current densities are the average current density (javg), the root mean square (RMS) current density (jrms), and the peak current density (jpeak). The peak current density (jpeak) is simply the current density corresponding to the peak current level of the waveform, where the waveform refers to the visual shape (amplitude vs. time) resulting from pulsed electrical current through a conductor:
                              j          peak                =                              I            peak                    A                                    (        1        )            where A is the cross-sectional area of the interconnect or other electrical conductor line. Equation 1 does not consider the width of the pulse or the adjacent layers (e.g., dielectric layers) to the electrical conductor line which determines the thermal resistance reflected in a thermal time constant (τ) associated with the conductor line. Designing ICs to have a jpeak at or below an allowed jpeak limit (or jpeak specification) avoids unwanted melting failure in the IC. For example, if a given design results in a metal interconnect line having a jpeak≧a Jpeak specification limit, the metal line widths may be increased to reduce the jpeak.
The average current density is defined as:
                              j          avg                =                              1            T                    ⁢                                    ∫              0              T                        ⁢                                          j                ⁡                                  (                  t                  )                                            ⁢                                                          ⁢                              ⅆ                t                                                                        (        2        )            where T is the time period of the current waveform. The RMS current density is defined over T as:
                              j          rms                =                                            1              T                        ⁢                                          ∫                0                T                            ⁢                                                                    j                    2                                    ⁡                                      (                    t                    )                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                                                                        (        3        )            